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Find the value of the following expression and round to the nearest integer: sum_(n=2)^(22)40(1.2)^(n-1) Answer:

Find the value of the following expression and round to the nearest integer:\newline∑n=22240(1.2)n−1 \sum_{n=2}^{22} 40(1.2)^{n-1} \newlineAnswer:

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Q. Find the value of the following expression and round to the nearest integer:\newline∑n=22240(1.2)n−1 \sum_{n=2}^{22} 40(1.2)^{n-1} \newlineAnswer:
  1. Recognize as geometric series: Recognize the sum as a geometric series. The general form of a geometric series is ∑n=0∞arn\sum_{n=0}^{\infty} ar^n, where aa is the first term and rr is the common ratio. In this case, a=40(1.2)2−1=40(1.2)a = 40(1.2)^{2-1} = 40(1.2) and r=1.2r = 1.2.
  2. Calculate first term: Calculate the first term of the series.\newlineThe first term is when n=2n=2, so we have a=40(1.2)2−1=40(1.2)=48a = 40(1.2)^{2-1} = 40(1.2) = 48.
  3. Use formula for sum: Use the formula for the sum of a finite geometric series.\newlineThe sum of the first kk terms of a geometric series is given by Sk=a(1−rk)(1−r)S_k = \frac{a(1 - r^k)}{(1 - r)}, where kk is the number of terms. Here, k=22−2+1=21k = 22 - 2 + 1 = 21 terms.
  4. Calculate sum: Calculate the sum of the series.\newlineUsing the formula from Step 33, we have S21=48(1−1.221)/(1−1.2)S_{21} = 48(1 - 1.2^{21}) / (1 - 1.2).
  5. Perform calculations: Perform the calculations.\newlineS21=48(1−1.221)1−1.2=48(1−1.221)−0.2=−240(1−1.221)S_{21} = \frac{48(1 - 1.2^{21})}{1 - 1.2} = \frac{48(1 - 1.2^{21})}{-0.2} = -240(1 - 1.2^{21}).
  6. Calculate exact value: Calculate the exact value of 1.2211.2^{21}. 1.2211.2^{21} is approximately 109.41898913151242109.41898913151242 (using a calculator).
  7. Substitute value into sum: Substitute the value from Step 66 into the sum.\newlineS21=−240(1−109.41898913151242)=−240(−108.41898913151242)=26020.55747150297S_{21} = -240(1 - 109.41898913151242) = -240(-108.41898913151242) = 26020.55747150297.
  8. Round to nearest integer: Round the result to the nearest integer.\newlineThe sum rounded to the nearest integer is approximately 2602126021.

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